Dynamic Foraging Patterns in Antarctic Humpbacks
Ben Weinstein
2017-01-05 15:23:37
## Source: local data frame [12 x 2]
##
## Animal max(timestamp, na.rm = T)
## (int) (time)
## 1 112699 2012-06-17 03:57:31
## 2 121207 2013-05-09 18:49:37
## 3 121208 2013-02-18 07:52:02
## 4 121210 2013-05-05 07:44:41
## 5 123224 2013-05-24 12:13:18
## 6 123232 2013-09-28 07:28:23
## 7 123236 2013-03-18 11:26:09
## 8 131127 2016-07-15 00:58:36
## 9 131130 2016-04-29 17:30:06
## 10 131132 2016-05-10 12:44:39
## 11 131133 2016-07-05 10:55:56
## 12 131136 2016-06-30 10:57:06
0.1 By Month
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
1.3.1 After filitering
How did the filter change the extent of tracks?
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha_mu[state[i,g,t-1],Month[i,g,t]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t],Month[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha_mu[state[i,g,steps[i,g]-1],Month[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
#for each month
for (m in 1:Months){
#Intercepts
alpha_mu[1,m] ~ dbeta(1,1)
alpha_mu[2,m] ~ dbeta(1,1)
gamma[1,m] ~ dbeta(3,2) ## gamma for state 1
dev[m] ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2,m] <- gamma[1,m] * dev[m]
}
##Behavioral States
#Hierarchical structure across motnhs
#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)sink()
## user system elapsed
## 376.629 2.228 31671.4611.4 Chains
## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1510007 80.7 3886542 207.6 3886542 207.6
## Vcells 303622995 2316.5 615471729 4695.7 524544016 4002.0## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1342652 71.8 3886542 207.6 3886542 207.6
## Vcells 46780009 357.0 492377383 3756.6 524544016 4002.0
1.5 Change in autocorrelation over time
2 Change in transition probabilities over time
2.1 Parameter Summary
## parameter par mean lower upper
## 1 alpha_mu alpha_mu[1,1] 0.80786562 0.727588008 0.88232371
## 2 alpha_mu alpha_mu[2,1] 0.17972076 0.102290383 0.27676781
## 3 alpha_mu alpha_mu[1,2] 0.87876765 0.788978069 0.94197718
## 4 alpha_mu alpha_mu[2,2] 0.12379987 0.062600037 0.22258545
## 5 alpha_mu alpha_mu[1,3] 0.87763694 0.790017951 0.94699432
## 6 alpha_mu alpha_mu[2,3] 0.07720100 0.037507166 0.12881625
## 7 alpha_mu alpha_mu[1,4] 0.79396827 0.597860382 0.96872722
## 8 alpha_mu alpha_mu[2,4] 0.13742905 0.018127646 0.27719349
## 9 alpha_mu alpha_mu[1,5] 0.77982923 0.600140591 0.92319216
## 10 alpha_mu alpha_mu[2,5] 0.08655756 0.040833160 0.15333219
## 11 alpha_mu alpha_mu[1,6] 0.70118590 0.439565389 0.89685092
## 12 alpha_mu alpha_mu[2,6] 0.26979798 0.025239661 0.77257307
## 13 gamma gamma[1,1] 0.92061890 0.873644343 0.96821525
## 14 gamma gamma[2,1] 0.17773231 0.035652292 0.32477815
## 15 gamma gamma[1,2] 0.81774154 0.762449776 0.88000820
## 16 gamma gamma[2,2] 0.09282394 0.006583999 0.21763372
## 17 gamma gamma[1,3] 0.77062799 0.683809107 0.85829430
## 18 gamma gamma[2,3] 0.08840825 0.005607181 0.21804059
## 19 gamma gamma[1,4] 0.84942208 0.716427984 0.96984570
## 20 gamma gamma[2,4] 0.11623297 0.006300703 0.32940711
## 21 gamma gamma[1,5] 0.72444491 0.574943397 0.88725878
## 22 gamma gamma[2,5] 0.41410676 0.228432706 0.58101952
## 23 gamma gamma[1,6] 0.87783287 0.744588038 0.97280848
## 24 gamma gamma[2,6] 0.33936652 0.047593343 0.67648245
## 25 theta theta[1] 0.02566977 0.008399520 0.04154484
## 26 theta theta[2] 3.11191766 2.835687052 3.33142447
3 Behavioral Prediction
3.1 Spatial Prediction
3.2 By individual
3.3 Autocorrelation in behavior
3.3.1 As single timeline
4 Simulated tracks
4.1 Behavioral description
5 Proportion of states by month
6 Distance between bouts
7 Behavior duration
## Source: local data frame [335 x 7]
## Groups: Animal, Track [103]
##
## Animal Track Bout MonthF phistate Days Month
## (dbl) (dbl) (int) (dbl) (fctr) (dbl) (fctr)
## 1 1 1 1 1 Traveling 14.541667 January
## 2 1 2 1 1 Traveling 11.889167 January
## 3 1 3 1 1 Traveling 15.561389 January
## 4 1 4 1 1 Traveling 22.269167 January
## 5 1 5 1 1 Traveling 15.600278 January
## 6 1 6 1 1 Area-restricted Search 2.603125 January
## 7 1 7 1 2 Traveling 23.679444 February
## 8 1 8 1 2 Traveling 13.104722 February
## 9 1 9 1 2 Area-restricted Search 3.349606 February
## 10 1 10 1 2 Traveling 22.118333 February
## .. ... ... ... ... ... ... ...
7.1 Proportion of time allocation
## Month Traveling Area-restricted Search PropF TotalTime
## 1 January 291.8011 242.26443 0.4536230 534.0655
## 2 February 447.4278 501.15394 0.5283192 948.5817
## 3 March 221.2866 208.51360 0.4851408 429.8002
## 4 April 162.5185 263.15661 0.6182100 425.6751
## 5 May 50.3569 119.85499 0.7041517 170.2119
## 6 June 127.9336 57.77661 0.3111116 185.71027.2 Number of bouts
8 Time spent in grid cell
8.1 All years
8.2 Add in sea ice
